Walsh Hadamard Transform

The Walsh Hadamard Transform.

What it does:

• The WHT decomposes a signal into its Walsh spectrum, revealing its frequency content in a different way than the Fourier Transform.

• The FWHT efficiently computes the WHT, significantly reducing the number of calculations compared to a naive implementation.

Why it's faster:

• A straightforward WHT takes O(N^2) operations for a data set of size N.

• FWHT brings this down to O(N log N) using a divide-and-conquer approach.

• It breaks down the WHT of size N into smaller WHTs of size N/2 and cleverly combines the results.

The inner workings (conceptually):

• FWHT relies on the structure of Hadamard matrices, which are filled with +1s and -1s.

• It leverages bitwise XOR operations to achieve efficient calculations.

• The algorithm resembles the Fast Fourier Transform (FFT) in its divide-and-conquer strategy and use of butterfly structures for computations.

Applications:

• FWHT finds use in various fields like:

  • Signal processing for feature extraction and analysis.

  • Error correction and coding.

  • Image processing applications like ghost imaging.

Further exploration:

• For a deeper understanding, you can explore resources on Hadamard matrices, bitwise XOR operations, and the connection between FWHT and FFT.

The first 16 Walsh functions. The number of each function is given in S, sequency order; D, dyadic order; and N, natural order. The n indices are also shown for the first five Rademacher functions, R; and DIF functions, O.